Receiver with colpitts differential oscillator, colpitts quadrature oscillator, and common-gate low noise amplifier

ABSTRACT

Embodiments of the present invention include a common-gate amplifier having an input terminal and an output terminal, a transistor having a source, a drain, and a gate, four inductors, and two capacitors, and a negative amplification circuitry. The negative amplification circuitry has an input terminal to receive an RF signal. The negative amplification circuitry applies negative or zero amplification to the RF signal and outputs the negative or zero amplified signal on an output terminal. Alternative embodiments include a Colpitts differential oscillator, which includes two Colpitts single-ended oscillators. Each Colpitts single-ended oscillator includes a transistor. The source of the transistor in one Colpitts single-ended oscillator may be capacitively coupled to the gate of the transistor in the other Colpitts single-ended oscillator.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a Divisional of U.S. application Ser. No.11/344,549, filed Jan. 30, 2006.

GOVERNMENT LICENSE RIGHTS

This invention was made with government support under Grant No.CCR-0086032 and Grant No. CCR-0120255, both awarded by the NationalScience Foundation. The government has certain rights in the invention.

BACKGROUND

1. Field

Embodiments of the present invention are related to communicationreceivers and, in particular, to receivers that includevoltage-controlled oscillators (VCO) and low-noise amplifiers (LNA).

2. Discussion of Related Art

Wireless communication systems commonly include cellular phones, radios,and radar systems, for example. A typical wireless system includes areceiver that receives a signal carried on an electromagnetic wave, suchas a radio frequency (RF) signal.

The ever-increasing interest in wireless communication systems also isemphasizing higher levels of integration, more complex functionalitiesand lower cost in integrated circuit (IC) implementations. Recently,complementary metal oxide semiconductor (CMOS) has emerged as viablealternative for RF and microwave integrated circuit (MIC) designs owingto the continued scaling of minimum feature size. The ability tointegrate complex digital signal processing functions makes CMOS anattractive candidate for system-on-chip (SOC) solutions. System-on-chipnecessitates the implementation of different analog building blocks onthe same die together with the large-scale digital circuits required bythe complex digital signal processing (DSP) functions. Unfortunately,parasitic coupling in silicon technology, especially through theconductive silicon substrate, makes it possible for digital switchingnoise to corrupt the weak analog and RF signals.

FIG. 1 is a simplified schematic diagram of a wireless receiver 100. Thereceiver 100 includes an antenna 102 coupled to a bandpass filter 104.The bandpass filter 104 is coupled to a low-noise amplifier (LNA) 106,which is coupled to a mixer 108. The mixer 108 is coupled to a low-passfilter (LPF) 110, which is coupled to an intermediate frequency (IF)amplifier 112. The IF amplifier 112 is coupled to an analog-to-digitalconverter (ADC) 114.

The receiver 100 may operate as follows. An RF signal may be transmittedto the receiver 100 and may arrive at the antenna 102. The bandpassfilter 104 may filter out all frequencies in the RF signal that do notfall within its pass band. The LNA 106 may amplify the filtered signaland suppresses noise contributed by the filter 104. The mixer 108 maydown-convert the filtered and amplified signals to a lower intermediatefrequency (IF). The low pass filter (LPF) 110 may low-pass the outputsignal from the mixer 108. The IF amplifier 112 may amplify the signalfrom the LPF 110. The ADC 114 may convert the output of the IF amplifier112 to a digital base band signal, such as to an audio frequency, forexample.

A role that the LNA 106 plays in the receiver 100 is appreciated byconsidering the Friis equation, which states that noise from blocksfollowing the LNA 106, typically the mixer 108, is suppressed by thegain of the LNA [1] [2] [3]. Therefore, low noise and high gain are themost important figures of merit of an LNA. In addition, LNA designinvolves tradeoffs between linearity, stability, input matching, andpower consumption.

Two popular topologies are widely used to implement a LNA such as theLNA 106, namely, the common-gate configuration and common-sourceconfiguration. Common-gate topology has such advantages as lower powerconsumption, easier input matching, less sensitivity to parasiticcapacitances, and better reverse isolation compared to those ofcommon-gate topology. However, the higher noise figure and lower gain ofthe common-gate configuration impede it from being widely used. Instead,a common-source amplifier with inductive degeneration is a more popularchoice in designing LNA circuits.

Phase-locked loops (PLL) are widely used in many applications, includingradio frequency synthesis, clock generation and clock and data recovery(CDR) circuits. Shown in FIG. 2 is a typical block diagram of a PLL 200in which the output ω_(LO) of the PLL 200 can be used as an input to themixer 108. The illustrated PLL 200 includes a reference frequency source202 coupled to a phase-frequency detector (PFD) 204, whose output iscoupled to a loop filter 206. The loop filter 206 is coupled to avoltage-controlled oscillator (VCO) 208, which is coupled to adivide-by-N circuit 210. The output of the divide-by-N circuit 210 iscoupled to a second input of the PFD 204.

In operation, the PFD 204 detects the phase and/or frequency differencebetween the reference frequency supplied by the reference frequencysource 202 and the output of divide-by-N circuit 210 and generates a DCvoltage proportional to the difference. The loop filter 206 attenuateshigh frequency components and extracts the DC information from the PFD204 output signal. The VCO 208 generates an oscillating waveform at thedesired frequency according to the voltage 212 output from the loopfilter 206.

In a PLL, all the constituent circuit blocks such as the VCO, frequencydivider, loop filter, charge pump, and PFD contribute phase noise at theoutput. Typically, it is often the case that the VCO is the dominantsource of phase noise. Therefore, extensive effort has been invested inimproving the phase noise performance of oscillators.

BRIEF DESCRIPTION OF THE DRAWINGS

In the drawings, like reference numbers generally indicate identical,functionally similar, and/or structurally equivalent elements. Thedrawing in which an element first appears may be indicated by theleftmost digit(s) in the reference number, in which:

FIG. 1 is a simplified schematic diagram of a wireless receiver;

FIG. 2 is a simplified block diagram of a phase-locked loop (PLL);

FIG. 3 is a schematic diagram of a general structure of a common-gateamplifier according to an embodiment of the present invention;

FIG. 4 is a graphical representation illustrating the comparison ofnoise figures between a conventional common-source low noise amplifier(CSLNA), a conventional common-gate low noise amplifier (CGLNA), and acommon-gate low noise amplifier (CGLNA) implemented according to anembodiment of the present invention;

FIG. 5 is a schematic diagram of a general structure of a single-endedimplementation of a common-gate amplifier according to an embodiment ofthe present invention;

FIG. 6 is a schematic diagram of a general structure of afully-differential implementation of a common-gate amplifier accordingto an embodiment of the present invention;

FIG. 7 is a graphical representation of measured and plotted Sparameters of the common-gate low noise amplifier (CGLNA) depicted inFIG. 3 and/or FIG. 5 according to an embodiment of the presentinvention;

FIG. 8 is a graphical representation of measured and plotted third orderinput referred intercept point (IIP3) and noise figure (NF) of thecommon-gate low noise amplifier (CGLNA) depicted in FIG. 3 and/or FIG. 5according to an embodiment of the present invention;

FIG. 9 is a schematic diagram of a Colpitts differential oscillatoraccording to an embodiment of the present invention;

FIG. 10 is a schematic diagram of a Colpitts differential oscillatoraccording to an alternative embodiment of the present invention;

FIG. 11 is a schematic diagram of a Colpitts differential oscillatoraccording to still another embodiment of the present invention;

FIG. 12 is a schematic diagram of a Colpitts quadrature oscillatoraccording to an embodiment of the present invention;

FIG. 13 is a graphical representation of measured and plotted tuningcurves for the Colpitts differential oscillator depicted in FIG. 10 andColpitts quadrature oscillator depicted in FIG. 12 according to anembodiment of the present invention; and

FIG. 14 is a graphical representation of measured and plotted phasenoises for the Colpitts differential oscillator depicted in FIG. 10 andColpitts quadrature oscillator depicted in FIG. 12 according to anembodiment of the present invention.

DETAILED DESCRIPTION OF THE ILLUSTRATED EMBODIMENTS

In a conventional common-gate LNA, noise factor is given by

$\begin{matrix}{F = {{1 + \frac{\overset{\_}{i_{nd}^{2}} \cdot \left( \frac{1}{1 + {g_{m}R_{S}}} \right)^{2}}{\overset{\_}{i_{nS}^{2}} \cdot \left( \frac{g_{m}R_{S}}{1 + {g_{m}R_{S}}} \right)^{2}}} = {1 + {\frac{\overset{\_}{i_{nd}^{2}}}{\overset{\_}{i_{nS}^{2}}}\left( \frac{1}{g_{m}R_{S}} \right)^{2}}}}} & (1)\end{matrix}$With i_(nd) ² =4kTγg_(d0)Δf, i_(nS) ² =4kTR_(S) ⁻¹Δf and α=g_(m)/g_(d0),(1) reduces to

$\begin{matrix}{F = {{1 + {\frac{4{kT}\;{\gamma g}_{d\; 0}\Delta\; f}{4{kTR}_{S}^{- 1}\Delta\; f}\left( \frac{1}{g_{m}R_{S}} \right)^{2}}} = {{1 + \frac{\gamma\; g_{d0}}{g_{m}^{2}R_{S}}} = \left. {1 + \frac{\gamma}{\alpha}} \right|_{{g_{m}R_{S}} = 1}}}} & (2)\end{matrix}$where α and γ are empirical process- and bias-dependent parameters,g_(m) is the effective small signal transconductance, R_(S) is the inputimpedance, F is the noise factor, g_(d0) is the drain-source conductanceat zero drain-source voltage, k is Boltzmann's constant, T is theabsolute temperature in Kelvins, i_(n)S is the noise current from sourceimpedance, and i_(nd) is the channel noise of MOSFET. Embodiments of thepresent invention are based on the following observation, that is, inorder to lower noise figure F, α in (2) should be effectively increased.This goal is met by increasing the effective small signaltransconductance g_(m) while keeping g_(d0) unchanged, as proposed inthe current invention.

FIG. 3 is a schematic diagram of the general structure of a common-gateamplifier 300 according to an embodiment of the present invention. Inthe illustrated embodiment, the common-gate amplifier 300 includes aninput node 306 and an output node Vout 304, a transistor 308, inductor312, inductor 310, a capacitor 314, and negative amplification circuitry302. The input of the negative amplification circuitry 302 is coupled tothe source of the transistor 308 and the input node 306. The output ofthe negative amplification circuitry 302 is coupled to the gate of thetransistor 308. The drain of the transistor 308 is coupled to oneterminal of the inductor 312. The other terminal of the inductor 312 iscoupled to a power supply Vdd 320. The capacitor 314 is coupled betweenthe drain of the transistor 308 and output node 304. The inductor 312and the capacitor 314 form a matching network, to match to 50Ω, forexample. The inductor 310 is coupled between the source of thetransistor 308 and ground 322. The inductor 310 provides high impedanceat the operating frequency of the common-gate amplifier 300.

In operation, an input voltage Vin is injected into the source of thetransistor 308. The voltage swing across gate-source capacitance of thetransistor 308 generates a small signal current flowing to the loadimpedance (i.e., the inductor 312, the capacitor 314, and the impedancelooking into drain of the transistor 308). The inductor 312 and thecapacitor 314 form the matching network which transforms the impedancelooking into drain of the transistor 308 to 50Ω for maximum power match.

In the illustrated embodiment, instead of shorting the gate terminal ofthe transistor 308 to AC ground, the negative amplification circuitry306 introduces a negative amplification between the source of thetransistor 308 and the gate of the transistor 308. The effectivetransconductance looking into the source terminal is changed fromG_(m)=g_(m), as in the conventional CGLNA, to G_(m)=(1+A)g_(m), where Ais the gain from source to gate. The resulting noise factor may becalculated to be

$\begin{matrix}{F = {{1 + {\frac{4{kT}\;\gamma\; g_{d\; 0}\Delta\; f}{4k\;{TR}_{S}^{- 1}\Delta\; f}\left( \frac{1}{\left( {1 + A} \right)g_{m}R_{S}} \right)^{2}}} = {{1 + \frac{\gamma\; g_{d0}}{\left( {1 + A} \right)^{2}g_{m}^{2}R_{S}}} = \left. {1 + \frac{\gamma}{\alpha\left( {1 + A} \right)}} \right|_{{{({1 + A})}g_{m}R_{S}} = 1}}}} & (3)\end{matrix}$

Comparing the noise factor for the conventional common-gate amplifier(shown in (2)) with respect to embodiments of the present invention(shown in (3)), it can be clearly seen that noise factor (F) is lowered.

In addition to a noise performance improvement, the common-gate lownoise amplifier implemented according to embodiments of the presentinvention may consume less power as compared to conventional CGLNA. Morespecifically, since (1+A)g_(m,new)=1/R_(S), g_(m,new) is lowered to1/(1+A)g_(m) and the DC power is lowered by the same amount.

FIG. 4 is a graphical representation 400 illustrating the comparison ofnoise figures between a conventional CSLNA (curve 402), a conventionalCGLNA (curve 404), and a common-gate low noise amplifier implementedaccording to embodiments of the present invention (e.g., the common-gateamplifier 300) (curve 406). It can be seen that the common-gate lownoise amplifier implemented according to embodiments of the presentinvention outperforms the conventional CSLNA when ω/ω_(T) is higher than0.35.

FIG. 5 is a schematic diagram of a single-ended common-gate amplifier500 according to an embodiment. In the illustrated embodiment, thecommon-gate amplifier 500 includes an input node Vin 502 and an outputnode Vout 522, a transistor 530, a transistor 532, an inductor 518, atransformer 506 with a primary inductor Lp and a secondary inductor Ls,a capacitor 520, a capacitor 524 and a capacitor 504. The input node 502is coupled to the transistor 530's source 508 through the couplingcapacitor 504. The transformer 506 couples the transistor 530's source508 and the transistor 530's gate 510 together. The transistor 530'sdrain 514 is coupled to the transistor 532's source. The transistor532's drain 516 is coupled to one terminal of the inductor 518. Theother terminal of the inductor 518 is coupled to Vdd 526. The capacitor520 is coupled between the transistor 532's drain 516 and the outputnode Vout 522. The capacitor 524 is coupled between output node Vout 522and ground 528.

The transistor 530 acts as the input transistor of common-gate topologyand is the main noise contributor. A DC bias voltage 512 provides ACground for the transformer 506's secondary inductor Ls and also the biasfor the transistor 530's gate 510. The transistor 532's gate is coupledto a second bias voltage 534.

In operation, the input voltage Vin is AC coupled through the couplingcapacitor 504 to the source of the transistor 530, which acts as theinput transistor of common-gate topology. The transformer 506 implementsthe negative amplification circuitry 302 in the general structure of thecommon-gate amplifier 300 (shown in FIG. 3) according to an embodimentof the present invention. The voltage swing across the gate-sourcecapacitance of the transistor 530 generates a small signal currentflowing to the source of the transistor 532. The transistor 532 acts ascascode transistor to suppress the Miller effect of the gate-draincapacitance of the transistor 530 and improves the reverse isolation andstability. The small signal coming out of the drain of the transistor532 flows to the load impedance (i.e., the inductor 518, the capacitor520, the capacitor 522, and the impedance looking into drain of thetransistor 532). The inductor 518, the capacitor C1 520, and thecapacitor 522 form the matching network which transforms the impedancelooking into drain of M2 532 to 50Ω for maximum power match.

As shown in FIG. 5, the use of an on-chip transformer 506 to implementcoupling of the source and gate terminals of the transistor 530 ensuresthe anti-phase operation between the source and the gate terminals. Theinverse gain

$A = {{kn} = {k\sqrt{\frac{L_{S}}{L_{P}}}}}$is determined by the turn ratio n and the coupling factor k. It can beshown that the small-signal input admittance at the source of thetransistor 530 is

$\begin{matrix}{Y_{in} \approx {\frac{1}{{sL}_{P}} + {\left( {1 + {nk}} \right)g_{m}} + {\left( {1 + {2{nk}} + n^{2}} \right){sC}_{gs}}}} & (4)\end{matrix}$

The above expressions clearly show that the transformer couplingeffectively increases the transconductance and therefore enables areduction in power consumption. The noise factor is reduced to1+γ/(1+A)α under the new input matching condition of (1+A)g_(m)R_(S)=1.

FIG. 6 is a schematic diagram of a fully-differential common-gateamplifier 600 according to an embodiment. It is the fully-differentialcounterpart of single-ended common-gate amplifier 500. In theillustrated embodiment, the amplifier 600 includes a positive input nodeVin+ 602 coupled to a first terminal of a capacitor 604. The secondterminal of the capacitor 604 is coupled to a transformer 606 comprisinga primary inductor Lp and a secondary inductor Ls. A second terminal ofthe capacitor 604 is coupled to the source of a transistor 608, whosedrain is coupled to the source of a transistor 610. The drain of thetransistor 610 is coupled to one terminal of a capacitor 612, whoseother terminal is coupled to a positive output node Vin+ 614. The otherterminal of the capacitor 612 is coupled to one terminal of an inductor616. The second terminal of the inductor 616 is coupled to Vdd 618,which is also coupled to one terminal of an inductor 620. The otherterminal of the inductor 620 is coupled to a negative output node Vin−622 via a capacitor 624. The other end of the inductor 620 also iscoupled to the drain of a transistor 626. The source of the transistor626 is coupled to the drain of a transistor 628, whose source is coupledto a negative input node Vin− 632 via a capacitor 630. A DC bias voltage636 provides AC ground for the transformer 606's secondary inductor Lsand also the bias for the gates of the transistors 608 and 628. Thegates of the transistor 610 and 626 are coupled to a second bias voltage634.

FIG. 7 is a graphical representation of measured and plotted Sparameters for a common-gate amplifier implemented according toembodiments of the present invention. The measured S-parameters areclose to simulated results in Spectre. Thanks to the wideband inputmatching of the common-gate configuration, S11 represented by curve 702remains less than −10 dB over the frequency range from 4 GHz to 7 GHz,which demonstrates that a common-gate amplifier implemented according toembodiments of the present invention can tolerate process, voltage, andtemperature variations as well as modeling errors. Compared to acommon-source amplifier, a common-gate amplifier implemented accordingto embodiments of the present invention may be more robust for inputmatching. S21 represented by curve 704 shows peak of 9.8 dB at 5.8 GHz.S22 represented by curve 706 is below −10 dB around 5.8 GHz. S12represented by curve 708 is below −30 dB from 4 GHz to 7 GHz.

FIG. 8 is a graphical representation of measured and plotted IIP3 curve802 and noise figure curve 804 for a common-gate amplifier implementedaccording to embodiments of the present invention. The measured IIP3 is7.6 dBm and noise figure is 2.5 dB at 5.8 GHz.

One common-gate amplifier implemented according to embodiments of thepresent invention consumes 1.9 mA from 1.8V. The measured IIP3 is 7.6dBm and the noise figure at maximum gain is 2.5 dB. Note that thecurrent consumption may be very low owing to the g_(m)-boosted effect.Note also that the common-gate amplifier implemented according toembodiments of the present invention also achieves very good linearity.A common-source configuration may achieve similar or a better noisefigure, but at the cost of a much higher power dissipation. Table 1below summarizes the measured performance of the common-gate amplifier500 according to an embodiment of the present invention

TABLE 1 Measured Performance Summary for the common-gate amplifier 400Technology 0.18 μm CMOS Power Supply 1.8 V DC Current 1.9 mA OperatingFrequency 5.8 GHz Noise Figure 2.5 dB IIP3 7.6 dBm S-Parameters S11−13.5 dB S21  +9.4 dB S12 −30.3 dB S22 −14.8 dB Die Area 910 μm × 670 μm

Embodiments of the present invention also include Colpitts oscillatorsthat may have improved start-up and phase noise. From the phase noise'spoint of view, Colpitts topology exhibits superior phase noisecharacteristic to that of cross-coupled topology. However, Colpittsoscillator requires more DC power consumption for reliable start-up andhas less tuning range than cross-coupled oscillator. To relax thestart-up issue in Colpitts oscillator, the gm-boosting techniqueimplemented in the embodiments of the present invention of gm-boostingcommon-gate amplifier can be applied to Colpitts topology as well. As aresult of gm-boosting, the effective small-signal negativetransconductance may be increased and the start-up condition may berelaxed.

FIG. 9 is a schematic diagram of a Colpitts differential oscillator 900according to an embodiment of the present invention. In the illustratedembodiment, the oscillator 900 includes a transistor 902, a transistor904, a capacitor 906, a capacitor 910, a capacitor 908, a capacitor 912,a capacitor 914, a capacitor 916, a current source 918, a second currentsource 920, a bias resistor 922, a bias resistor 924, a bias voltageV_(B) 926, an inductor 928, an inductor 930, a positive output node V+932 and a negative output node V− 934.

In the Colpitts differential oscillator 900, the current source 918 iscoupled between the source of the transistor 902 and ground 942. Thecurrent source 920 is coupled between M2's source 938 and ground 942.The capacitor 914 is coupled between the source of the transistor 902and the gate of the transistor 904. The capacitor 916 is coupled betweenthe source of the transistor 902 and the gate of the transistor 904 s.The capacitor 909 is coupled between the positive output node V+ 932 andthe source of the transistor 902. The capacitor 910 is coupled betweennegative output node V− 934 and M2's source 938. The capacitor 908 iscoupled between the source of the transistor 902 and ground 942. Thecapacitor 912 is coupled between the source of the transistor 904 andground 942. The resistor 922 is coupled between the gate of thetransistor 902 and the bias voltage V_(B) 926. Resistor RB2 924 iscoupled between M2's gate 946 and the bias voltage V_(B) 926. Theinductor 928 is coupled between the positive output node V+ 932 and thepower supply Vdd 940. The inductor 930 is coupled between the negativeoutput node V− 934 and the power supply Vdd 940.

In the Colpitts differential oscillator 900, the current source 918provides the DC bias current for the transistor 902. The current source920 provides the DC bias current for the transistor 904. The capacitors906 and 908 provide the positive feedback from the drain of thetransistor 902 to the source of the transistor 902. The capacitors 910and 912 provide the positive feedback from the drain of the transistor904 to the source of the transistor 904. The capacitor 914 couples thesource of the transistor 902 to the gate of the transistor 904 andeffectively double the small signal transconductance of transistor 902.The capacitor 916 couples the source of the transistor 904 to the gateof the transistor 902 and effectively doubles the small signaltransconductance of the transistor 904. The resistor 922 provides the DCbias for the gate of the transistor 902. The resistor 924 provides theDC bias for the gate of the transistor 904. Due to the gm-boostingeffect, the start-up condition of the Colpitts differential oscillator900 may be relaxed to

$\begin{matrix}{{g_{m}R_{P}} > \frac{\left( {C_{1} + C_{2}} \right)^{2}}{C_{1}C_{2}} \geq 2} & (5)\end{matrix}$

FIG. 10 is a schematic diagram of a Colpitts differential oscillator1000 according to an alternative embodiment of the present invention. Inthe illustrated embodiment, the oscillator 1000 includes a transistor1002, a transistor 1004, a capacitor 1006, a capacitor 1008, a capacitor1010, a capacitor 1012, a current source 1014, a second current source1016, an inductor 1018, an inductor 1020, a varactor 1022, a secondvaractor 1024, a positive output node V+ 1026 and a negative output nodeV− 1028.

In the Colpitts differential oscillator 1000, the current source 1014 iscoupled between the source of the transistor 1002 and ground 1038. Thecurrent source 1016 is coupled between the source of the transistor 1004and ground 1038. The capacitor 1006 is coupled between the positiveoutput node V+ 1026 and the source of the transistor 1002. The capacitor1008 is coupled between the negative output node V− 1028 and the sourceof the transistor 1004. The capacitor 1010 is coupled between the sourceof the transistor 1002 and ground 1038. The capacitor 1012 is coupledbetween the source of the transistor 1004 and ground 1038.

The gate of the transistor 1002 is coupled to the negative output nodeV− 1028. The gate of the transistor 1004 is coupled to the positiveoutput node V+ 1026. The drain of the transistor 1002 is coupled to thepositive output node V+ 1026. The drain of the transistor 1004 iscoupled to the negative output node V− 1028. The inductor 1018 iscoupled between the positive output node V+ 1026 and the power supplyVdd 1034. The inductor 1020 is coupled between the negative output nodeV− 1028 and the power supply Vdd 1034. The varactor 1022 is coupledbetween the positive output node V+ 1026 and the control voltage Vcrtl1036. The varactor 1024 is coupled between the negative output node V−1028 and the control voltage Vcrtl 1036.

In the Colpitts differential oscillator 1000, the inductor 1018, thecapacitor 1006, the capacitor 1010, and the varactor 1022 form the LCtank on the left side for oscillation. Similarly, the inductor 1020, thecapacitor 1008, the capacitor 1012, and the varactor 1024 form the LCtank on the right side for oscillation. The varactors 1022 and 1024 arevariable capacitors that implement the frequency tuning using thecontrol voltage 1036. The capacitors 1006 and 1010 provide the positivefeedback from the drain of the transistor 1002 to the source of thetransistor 1002. The positive feedback is used to provide negativeconductance to compensate the energy loss of the LC tank duringoscillation cycle and sustain the oscillation. Similarly, the capacitors1008 and 1012 provide the positive feedback from the drain of thetransistor 1004 to the source of the transistor 1004.

The drains and gates of the transistors 1002 and 1004's are crosscoupled. The cross-coupling between drains and gates of the transistors1002 and 1004's help to boost the small signal transconductance oftransistors 1002 and 1004 to higher value and therefore may result inmore relaxed start-up condition. The current source 1014 provides the DCbias current for the transistor 1002. The current source 1016 providesthe DC bias current for the transistor 1004. The positive output of theColpitts differential oscillator 1000 is at the drain of the transistor1026 and the negative output of the Colpitts differential oscillator1000 is at the drain of the transistor 1028.

In the Colpitts differential oscillator 1000, the current source 1014provides the DC bias current for the transistor 1002. The current source1016 provides the DC bias current for the transistor 1004. Thecapacitors 1006 and 1010 provide the positive feedback from the drain ofthe transistor 1002 to the source of the transistor 1002. The capacitors1008 and 1012 provide the positive feedback from the drain of thetransistor 1004 to the source of the transistor 1004. It can be shownthat the start-up condition of the Colpitts differential oscillator 1000is relaxed to

$\begin{matrix}{{g_{m}R_{P}} > \frac{2\left( {C_{1} + C_{2}} \right)^{2}}{C_{2}\left( {{2C_{1}} + C_{2}} \right)}} & (6)\end{matrix}$

The direct cross-coupling between the drains and gates of thetransistors 1002 and 1004 help to boost the small signaltransconductance of the transistors 1002 and 1004 to higher value andtherefore may result in more relaxed start-up condition, as implied by(5) and (6).

FIG. 11 is a schematic diagram of a Colpitts differential oscillator1100 according to an alternative embodiment of the present invention. Inthe illustrated embodiment, the oscillator 1100 includes a transistor1102, a transistor 1104, a transistor 1114, a transistor 1116, acapacitor 1106, a capacitor 1108, a capacitor 1110, a capacitor 1112, acurrent source 1140, an inductor 1118, an inductor 1120, a varactor1122, a varactor 1124, a positive output node V+ 1126 and a negativeoutput node V-1128.

In the Colpitts differential oscillator 1100, the current source 1140 iscoupled between the common sources of the transistors 1114 and 1116 andground 1138. The capacitor 1106 is coupled between the positive outputnode V+ 1126 and the source of the transistor 1102. The capacitor 1108is coupled between the negative output node V-1128 and the source of thetransistor 1104. The capacitor 1110 is coupled between the source of thetransistor 1102 and ground 1138. The capacitor 1112 is coupled betweenthe source of the transistor 1104 and ground 1138.

The gate of the transistor 1102 is coupled to the negative output nodeV-1128. Transistor M2's gate is coupled to the positive output node V+1126. The drain of the transistor 1102 is coupled to the positive outputnode V+ 1126. The drain of the transistor 1104 is coupled to thenegative output node V-1128. The drain of the transistor 1114 is coupledto the source of the transistor 1102. The drain of the transistor 1116is coupled to the source of the transistor 1104. The sources of thetransistors 1114 and 1116 are coupled together at one terminal ofcurrent source 1140. The gate of the transistor 1114 is coupled to thenegative output node V− 1128. The gate of the transistor 1116 is coupledto the positive output node V+ 1126. The inductor 1118 is coupledbetween the positive output node V+ 1126 and the power supply Vdd 1134.The inductor 1120 is coupled between the negative output node V-1128 andthe power supply Vdd 1134. The varactor 1122 is coupled between thepositive output node V+ 1126 and the control voltage Vcrtl 1136. Thevaractor 1124 is coupled between the negative output node V-1128 and thecontrol voltage Vcrtl 1136.

In the Colpitts differential oscillator 1100, the inductor 1118, thecapacitor 1106, the capacitor 1110, and the varactor 1122 form the LCtank on the left side for oscillation. Similarly, the inductor 1120, thecapacitor 1108, the capacitor 1112, and the varactor 1124 form the LCtank on the right side for oscillation. The varactors 1122 and 1124 arevariable capacitors that implement the frequency tuning. The capacitors1106 and 1110 provide the positive feedback from the drain of thetransistor 1102 to the source of the transistor 1102. The positivefeedback is used to provide negative conductance to compensate theenergy loss of the LC tank during oscillation cycle and sustain theoscillation. Similarly, the capacitors 1108 and 1112 provide thepositive feedback from the drain of the transistor 1004 to the source ofthe transistor 1004. The drains and gates of the transistors 1102 and1104 are directly cross coupled.

The direct cross-coupling between the drains and gates of thetransistors 1102 and 1104 help to boost the small signaltransconductance of the transistors 1102 and 1104 to higher value andtherefore may results in more relaxed start-up condition. The transistor1114 is coupled between the current source 1140 and the transistor 1102.Similarly, the transistor 1116 is coupled between the current source1140 and the transistor 1104. The current source 1140 provides the DCbias current for the transistors 1102, 1104, 1114 and 1116. The positiveoutput is at the drain of the transistor 1126 and the negative output isat the drain of the transistor 1128.

In the Colpitts differential oscillator 1100, the current source 1140provides the DC bias current for the transistors 1102 and 1104. Thecapacitors 1106 and 1110 provide the positive feedback from the drain ofthe transistor 1102 to the source of the transistor 1102. The capacitors1108 and 1112 provide the positive feedback from the drain of thetransistor 1104 to the source of the transistor 1104. The varactors 1122and 1124 provide means of frequency tuning.

The main difference between Colpitts differential oscillator 1100 andColpitts differential oscillator 1000 is that in the oscillator 1100,the transistors 1114 and 1116 are added to combine the current sources1014 and 1016 into one current source 1140. Simulations in CadenceSpectre show that this arrangement improves the far-out phase noiseperformance of the Colpitts differential oscillator 1100 as compared tothat of Colpitts differential oscillator 1000.

The embodiments implemented in FIG. 9, FIG. 10, and FIG. 11 provide thewelcomed differential outputs V+ and V−. Also, the anti-phaserelationship between the source and the gate of the cross-coupledtransistors 902/904, 1002/1004, and 1102/1104 may result in sharpercommutation and help to suppress the noise from differential pair [4].Therefore, better phase noise may be achieved.

Generating quadrature signals may be often required in RF application,such as in the direct conversion receiver (DCR) 100 depicted in FIG. 1,for example. This be can be done through the use of a quadraturevoltage-controlled oscillator (VCO), which is essentially a pair ofoscillators coupled to each other so that the voltage-controlledoscillator outputs quadrature signals directly [5].

FIG. 12 is a schematic diagram of a Colpitts quadrature oscillator 1200according to an embodiment of the present invention. The oscillator 1200may be based on the Colpitts differential oscillator 1100. In theillustrated embodiment, the quadrature oscillator 1200 includes atransistor 1202, a transistor 1204, a transistor 1206, a transistor1208, a transistor 1210, a transistor 1212, a transistor 1214, atransistor 1216, a capacitor 1218, a capacitor 1220, a capacitor 1222, acapacitor 1224, a capacitor 1226, a capacitor 1228, a capacitor 1230, acapacitor 1232, an in-phase positive output node I+ 1234, an in-phasenegative output node I− 1236, a quadrature positive output node Q+ 1238,a quadrature negative output node Q− 1240, a current source 1242, acurrent source 1244, an inductor 1246, an inductor 1248, an inductor1250, an inductor 1252, a varactor 1266, a varactor 1268, a varactor1270, a varactor 1272.

In the Colpitts quadrature oscillator 1200, the current source 1242 iscoupled between the common sources of the transistors 1206 and 1208 andground 1276. The current source 1244 is coupled between the commonsources of the transistors 1214 and 1216 and ground 1276. The capacitor1218 is coupled between the in-phase positive output node I+ 1234 andthe source of the transistor 1202. The capacitor 1220 is coupled betweenthe in-phase negative output node I− 1236 and the source of thetransistor 1204s. The capacitor 1222 is coupled between the source ofthe transistor 1202 and ground 1276. The capacitor 1224 is coupledbetween the source of the transistor 1204 and ground 1276. The capacitor1226 is coupled between the quadrature positive output node Q+ 1238 andthe source of the transistor 1210. The capacitor 1228 is coupled betweenthe quadrature negative output node Q− 1240 and the source of thetransistor 1212. The capacitor 1230 is coupled between the source of thetransistor 1210 and ground 1276. The capacitor 1232 is coupled betweenthe source of the transistor 1212 and ground 1276.

The gate of the transistor 1202 is coupled to the in-phase negativeoutput node I− 1236. The gate of the transistor 1204 is coupled to thein-phase positive output node I+ 1234. The drain of the transistor 1202is coupled to the in-phase positive output node I+ 1234. The drain ofthe transistor 1204 is coupled to the in-phase negative output node I−1236. The drain of the transistor 1206 is coupled to the source of thetransistor 1202. The drain of the transistor 1208 is coupled to thesource of the transistor 1204. The sources of the transistors 1206 and1208 are coupled together at one terminal of current source 1242. Thegate of the transistor 1206 is coupled to the quadrature positive outputnode Q+ 1238. The gate of the transistor 1208 is coupled to thequadrature negative output node Q− 1240. The gate of the transistor 1210is coupled to the quadrature negative output node Q− 1240. The gate ofthe transistor 1204 is coupled to the quadrature positive output node Q+1238. The drain of the transistor 1210 is coupled to the quadraturepositive output node Q+ 1238. The drain of the transistor 1212 iscoupled to the quadrature negative output node Q− 1240. The drain of thetransistor 1214 is coupled to the source of the transistor 1210. Thedrain of the transistor 1218 is coupled to the source of the transistor1212. The sources of the transistor 1214 and 1216 are coupled at oneterminal of current source 1244. The gate of transistor 1214 is coupledto the in-phase negative output node I− 1236. The gate of the transistor1216 is coupled to the in-phase positive output node I+ 1234.

The inductor 1246 is coupled between the in-phase positive output nodeI+ 1234 and the power supply 1274. The inductor 1248 is coupled betweenthe in-phase negative output node I− 1236 and the power supply 1274. Theinductor 1250 is coupled between the quadrature positive output node Q+1238 and the power supply 1274. The inductor 1252 is coupled between thequadrature negative output node Q− 1240 and the power supply 1274.

The varactor 1266 is coupled between the in-phase positive output nodeI+ 1234 and the control voltage Vcrtl 1278. The varactor 1268 is coupledbetween the in-phase negative output node I− 1236 and the controlvoltage Vcrtl 1278. The varactor 1270 is coupled between the quadraturepositive output node Q+ 1238 and the control voltage Vcrtl 1278. Thevaractor 1272 is coupled between the quadrature negative output node Q−1240 and the control voltage Vcrtl 1278.

In the Colpitts quadrature oscillator 1200, the inductor 1246, thecapacitor 1218, the capacitor 1222, and the varactor 1266 form the LCtank on the left side for the I oscillator. The inductor 1248, thecapacitor 1220, the capacitor 1224, and the varactor 1268 form the LCtank on the right side for the I oscillator. Similarly, the inductor1250, the capacitor 1226, the capacitor 1230, and the varactor 1270 formthe LC tank on the left side for the Q oscillator. The inductor 1252,the capacitor 1228, the capacitor 1232, and the varactor 1272 form theLC tank on the right side for the Q oscillator.

The varactors 1266, 1268, 1270, and 1272 are variable capacitors thatimplement the frequency tuning. The capacitors 1218 and 1222 provide thepositive feedback from the drain of the transistor 1202 to the source ofthe transistor 1202. The positive feedback is used to provide negativeconductance to compensate the energy loss of the LC tank duringoscillation cycle and sustain the oscillation. Similarly, the capacitors1220 and 1224 provide the positive feedback from the drain of thetransistor 1204 to the source of the transistor 1204. The capacitors1226 and 1230 provide the positive feedback from the drain of thetransistor 1210 to the source of the transistor 1210. The capacitors1228 and 1232 provide the positive feedback from the drain of thetransistor 1212 to the source of the transistor 1212.

The drains and gates of the transistors 1202 and 1204 are cross coupled.The cross-coupling between drains and gates of the transistors 1202 and1204 help to boost the small signal transconductance of the transistors1202 and 1204 to higher value and therefore may result in more relaxedstart-up condition. Similarly, the drains and gates of the transistors1210 and 1212 are directly cross coupled. The transistor 1206 is coupledbetween the current source 1242 and the transistor 1202. The transistor1208 is coupled between the current source 1242 and the transistor 1204.

The transistors 1206 and 1208 act as the coupling devices between Ioscillator and Q oscillator. Similarly, the transistor 1214 is coupledbetween the current source 1244 and the transistor 1210. The transistor1216 is coupled between the current source 1244 and the transistor 1212.Note that the couplings are in an inverse sense. The current source 1242provides the DC bias current for the transistors 1202, 1204, 1206, and1208. Similarly, the current source 1244 provides the DC bias currentfor the transistors 1210, 1212, 1214, and 1216. The positive output ofthe I oscillator is at the drain of the transistor 1202 and the negativeoutput of the I oscillator is at the drain of the transistor 1204.Similarly, the positive output of Q oscillator is at the drain of thetransistor 1210 and the negative output of Q oscillator is at the drainof the transistor 1212.

In the Colpitts quadrature oscillator 1200, the current source 1242provides the DC bias current for the transistors 1202 and 1204. Thecurrent source 1244 provides the DC bias current for the transistors1210 and 1212. The capacitors 1218 and 1222 provide the positivefeedback from the drain of the transistor 1202 to the source of thetransistor 1202. The capacitors 1220 and 1224 provide the positivefeedback from the drain of the transistor 1204 to the source of thetransistor 1204. The capacitors 1226 and 1230 provide the positivefeedback from the drain of the transistor 1210 to the source of thetransistor 1210. The capacitors 1228 and 1232 provide the positivefeedback from the drain of the transistor 1212 to the source of thetransistor 1204. The transistor 1206, 1208, 1214, and 1216 provide thecoupling between the in-phase oscillator and the quadrature oscillator.Note that the coupling is in a inverse sense. The varactors 1266, CV21268, 1270, and 1272 provide means of frequency tuning.

Simulations in Spectre show that the proposed Colpitts differentialoscillators 900/1000/1100 and Colpitts quadrature oscillator 1200achieve excellent phase noise performance. Compared to the standardcross-coupled oscillator, an average of 2 dB improvement at 3 MHz offsetand 6 dB improvement at close-in offset frequency are obtained.

To further evaluate the invention, Colpitts differential oscillator 1000and Colpitts quadrature oscillator 1200 are fabricated in a standard6-metal 0.18 μm CMOS RF process. FIG. 13 is a graphical representation1300 illustrating a plot of the tuning curve 1302 for the Colpittsdifferential oscillator 1000 and the tuning curve 1304 for the Colpittsquadrature oscillator 1200 according to an embodiment of the presentinvention. In the illustrated embodiment, the Colpitts differentialoscillator 1000 operates from 1.59 GHz to 1.98 GHz with a tuning rangeof 22%. The Colpitts quadrature oscillator 1100 operates from 1.83 GHzto 2.24 GHz corresponding to a 20% tuning range, which may be enough inreality considering the process, voltage and temperature (PVT)variations.

FIG. 14 is a graphical representation 1400 illustrating the measuredphase noise curve 1402 of a Colpitts LC oscillator 1000 according to anembodiment of the present invention and the measured phase noise curve1404 of the Colpitts quadrature oscillator 1200 according to anembodiment of the present invention.

Table 2 below summarizes the performance of the Colpitts differentialoscillator 1000 according to an embodiment of the present invention.Table 3 below lists the performance of the Colpitts quadratureoscillator 1200 according to an embodiment of the present invention. Therelatively small current consumption also validates the improvement ofeasier start-up.

TABLE 2 Measured Performance Summary for the Colpitts differentialoscillator 1000 Technology 0.18 μm CMOS Power Supply 2.0 V DC Current3.6 mA Frequency 1.59 GHz~1.98 GHz Tuning Range 22% Figure of MeritOffset Frequency Phase Noise (FOM) Phase Noise  @ 10 KHz  −87.7 dBc/Hz183.2 dB @ 100 KHz −105.8 dBc/Hz 181.3 dB @ 600 KHz −122.9 dBc/Hz 182.8dB   @ 1 MHz −128.0 dBc/Hz 183.5 dB Die Area 970 μm × 710 μm

TABLE 3 Measured Performance Summary for the Colpitts quadratureoscillator 1200 Technology 0.18 μm CMOS Power Supply 2.0 V DC Current4.3 mA Frequency 1.83 GHz~2.24 GHz Tuning Range 20% Figure of MeritOffset Frequency Phase Noise (FOM) Phase Noise  @ 10 KHz  −95.9 dBc/Hz191.8 dB @ 100 KHz −106.3 dBc/Hz 182.2 dB @ 600 KHz −121.4 dBc/Hz 181.7dB   @ 1 MHz −127.0 dBc/Hz 182.9 dB Die Area 1450 μm × 940 μm

As described above, embodiments of the present invention may beimplemented using hardware, software, or a combination thereof. Inimplementations using software, the software may be stored on amachine-accessible medium. A machine-accessible medium includes anymechanism that may be adapted to store and/or transmit information in aform accessible by a machine (e.g., a computer, network device, personaldigital assistant, manufacturing tool, any device with a set of one ormore processors, etc.). For example, a machine-accessible mediumincludes recordable and non-recordable media (e.g., read only memory(ROM), random access memory (RAM), magnetic disk storage media, opticalstorage media, flash memory devices, etc.), as recess as electrical,optical, acoustic, or other form of propagated signals (e.g., carrierwaves, infrared signals, digital signals, etc.).

Reference throughout this specification to “one embodiment” or “anembodiment” means that a particular feature, structure, process, block,or characteristic described in connection with an embodiment may beincluded in at least one embodiment of the present invention. Thus, theappearance of the phrases “in one embodiment” or “in an embodiment” invarious places throughout this be specification does not necessarilymean that the phrases all refer to the same embodiment. The particularfeatures, structures, or characteristics may be combined in any suitablemanner in one or more embodiments.

REFERENCES

[1] David J. Allstot, Short Course: Low Noise Amplifiers, IEEEInternational Solid State Circuit Conference, 2001.

[2] Razavi, R F Microelectronics, Prentice Hall, 1998.

[3] Thomas H. Lee, The Design of CMOS Radio Frequency IntegratedCircuits, Cambridge, 1998.

[4] A. Hajimiri, et al., “A General Theory of Phase Noise in ElectricalOscillators,” IEEE J. Solid-State Circuits, vol. 33, pp. 179-194,February 1998.

[5] A. Rofougaran, J. Rael, M. Rofougaran and A. A. Abidi, “A 900 MHzCMOS LC-oscillator with quadrature outputs,” IEEE InternationalSolid-State Circuits Conference, 1996, pp. 392-393.

1. An apparatus, comprising: a Colpitts differential oscillator having:a first Colpitts single-ended oscillator, the first Colpittssingle-ended oscillator having a first transistor, the first transistorhaving a source and a gate; and a second Colpitts single-endedoscillator, the second Colpitts single-ended oscillator having a secondtransistor, the second transistor having a source and a gate, whereinthe source of the first transistor is capacitively coupled to the gateof the second transistor, and wherein the source of the secondtransistor capacitively coupled to the gate of the first transistor. 2.The apparatus of claim 1, wherein the first Colpitts single-endedoscillator further comprises: a first terminal of a first inductorcoupled to V_(DD); a second terminal of the first inductor coupled tothe drain of the first transistor, to the first terminal of a firstcapacitor, and to a first output of the Colpitts differentialoscillator; a second terminal of first capacitor coupled to source ofthe first transistor, to the first terminal of a second capacitor, andto the first terminal of a third capacitor, the third capacitor tocapacitively couple the source of the first transistor to the gate ofthe second transistor; and a second terminal of the second capacitorcoupled to V_(SS).
 3. The apparatus of claim 2, wherein a secondterminal of the third capacitor is coupled to the gate of the secondtransistor.
 4. The apparatus of claim 1, wherein the second Colpittssingle-ended oscillator further comprises: a first terminal of a secondinductor coupled to V_(DD); a second terminal of the second inductorcoupled to the drain of the second transistor, to the first terminal ofa fourth capacitor, and to a second output of the Colpitts differentialoscillator; a second terminal of the fourth capacitor being coupled tothe source of the second transistor, to the first terminal of a fifthcapacitor, and to the first terminal of a sixth capacitor, the sixthcapacitor to capacitively couple the source of the second transistor tothe gate of the first transistor; and a second terminal of fifthcapacitor is coupled to V_(SS).
 5. The apparatus of claim 4, wherein asecond terminal of the sixth capacitor is coupled to the gate of thefirst transistor.
 6. The apparatus of claim 5, wherein a first terminalof a first resistor is coupled to the gate of the first transistor and asecond terminal of the first resistor is coupled to a bias voltage. 7.The apparatus of claim 6, wherein a first terminal of a second resistoris coupled to the gate of the second transistor and a second terminal ofthe second resistor is coupled to the bias voltage.
 8. An apparatus,comprising: a Colpitts differential oscillator having: a first Colpittssingle-ended oscillator, the first single-ended Colpitts oscillatorhaving a first transistor, the first transistor having a gate and adrain; and a second Colpitts single-ended oscillator, the secondColpitts single-ended oscillator having a second transistor, the secondtransistor having a gate and a drain, wherein the gate of the firsttransistor is coupled to the drain of the second transistor, and whereinthe gate of the second transistor is coupled to the drain of the firsttransistor.
 9. The apparatus of claim 8, wherein the first Colpittssingle-ended oscillator further comprises: a first terminal of a firstinductor coupled to V_(DD); a second terminal of the first inductorcoupled to the drain of the first transistor, to the first terminal of afirst capacitor, and to a first output of the Colpitts differentialoscillator; a second terminal of the first capacitor coupled to sourceof the first transistor and to the first terminal of a second capacitor;and a second terminal of the second capacitor is coupled to V_(SS). 10.The apparatus of claim 9, wherein the second Colpitts single-endedoscillator further comprises: a first terminal of a second inductorcoupled to V_(DD); a second terminal of the second inductor coupled tothe drain of the second transistor, to the first terminal of a thirdcapacitor, and to a second output of the Colpitts differentialoscillator; a second terminal of the third capacitor coupled to thesource of the second transistor and to a first terminal of fourthcapacitor, wherein a second terminal of the fourth capacitor is coupledto V_(SS).
 11. The apparatus of claim 10, further comprising: a firstvaractor; and a second varactor, wherein the first varactor includes afirst terminal coupled to the gate of the second transistor, the secondterminal of the first inductor, and to the first output for the firstColpitts differential oscillator, the first varactor having a secondterminal coupled to a control voltage and to a first terminal of thesecond varactor, wherein the second varactor further includes a secondterminal coupled to the gate of the first transistor, the secondterminal of the second inductor, and to the second output for the firstColpitts differential oscillator.
 12. The apparatus of claim 8, furthercomprising: a first current source to provide a DC bias current to thefirst transistor; and a second current source to provide a DC biascurrent to the second transistor.
 13. The apparatus of claim 8, furthercomprising a current source to provide a DC bias current to the firsttransistor and the second transistor.
 14. An apparatus, comprising: aColpitts quadrature oscillator having: a first Colpitts differentialoscillator having a first output, the first Colpitts differentialoscillator including a first Colpitts single-ended oscillator having afirst transistor and a second Colpitts single-ended oscillator having athird transistor, wherein the gate of the first transistor is coupled tothe drain of the third transistor and the gate of the third transistoris coupled to the drain of the first transistor; and a second Colpittsdifferential oscillator having a second output, wherein the first outputis coupled to the second Colpitts differential oscillator, and whereinthe second output is coupled to the first Colpitts differentialoscillator.
 15. The apparatus of claim 14, wherein: the first Colpittssingle-ended oscillator further comprises a second transistor coupled tothe first transistor; the second Colpitts single-ended oscillatorfurther comprises a fourth transistor coupled to the third transistor;and the second Colpitts differential oscillator comprises: a thirdColpitts single-ended oscillator having a fifth transistor and a sixthtransistor; and a fourth Colpitts single-ended oscillator having aseventh transistor and an eighth transistor, wherein a positive of thefirst output of the first Colpitts differential oscillator is coupled tothe gate of the eighth transistor, wherein a negative of the firstoutput of the first Colpitts differential oscillator is coupled to thegate of the sixth transistor, wherein a positive of the second output ofthe second Colpitts differential oscillator is coupled to the gate ofthe second transistor, and wherein a negative of the second output ofthe second Colpitts differential oscillator is coupled to the gate ofthe fourth transistor.
 16. The apparatus of claim 15, wherein the firstColpitts single-ended oscillator further comprises: a first terminal offirst inductor coupled to V_(DD); a second terminal of the firstinductor coupled to the drain of the first transistor, to the firstterminal of a first capacitor, to the positive of the first output ofthe first Colpitts differential oscillator, and to a source of thesecond transistor; a second terminal of the first capacitor coupled tosource of the first transistor, to the drain of the second transistor,and to a first terminal of a second transistor; and a second terminal ofthe second capacitor and the source of the second transistor coupled toV_(SS).
 17. The apparatus of claim 16, wherein the second Colpittssingle-ended oscillator further comprises: a first terminal of a secondinductor coupled to V_(DD); a second terminal of a second inductorcoupled to the drain of the third transistor, to a first terminal of athird capacitor, and to the negative of the first output of the firstColpitts differential oscillator; and a second terminal of a fourthcapacitor coupled to the source of the third transistor, to the drain oftransistor the fourth transistor, and to a first terminal of a fifthcapacitor, wherein a second terminal of the fifth capacitor is coupledto V_(SS).
 18. The apparatus of claim 15, wherein the third Colpittssingle-ended oscillator further comprises: a first terminal of a thirdinductor coupled to V_(DD); a second terminal of the third inductorcoupled to the drain of the fifth transistor, to the first terminal of asixth capacitor, to the positive of the first output of the secondColpitts differential oscillator, and to a gate of the seventhtransistor; a second terminal of the sixth capacitor coupled to sourceof the fifth transistor, to the drain of the sixth transistor, and to afirst terminal of a seventh capacitor; and a second terminal of theseventh capacitor and the source of the sixth transistor coupled toV_(SS).
 19. The apparatus of claim 15, wherein the fourth Colpittssingle-ended oscillator further comprises: a first terminal of a fourthinductor coupled to V_(DD); a second terminal of the fourth inductorcoupled to the drain of the seventh transistor, to a first terminal of aseventh capacitor, and to the negative of the first output of the secondColpitts differential oscillator; and a second terminal of the seventhcapacitor coupled to the source of the seventh transistor, to the drainof the eighth transistor, and to a first terminal of an eighthcapacitor, wherein a second terminal of the eighth capacitor is coupledto V_(SS).
 20. The apparatus of claim 15, wherein the gate of the fifthtransistor is coupled to the drain of the seventh transistor and whereinthe gate of the seventh transistor is coupled to the drain of the fifthtransistor.